Question: Simplify and expand the following expression: $ \dfrac{2}{y + 8}- \dfrac{3}{3y + 27}- \dfrac{1}{y^2 + 17y + 72} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the second term: $ \dfrac{3}{3y + 27} = \dfrac{3}{3(y + 9)}$ We can factor the quadratic in the third term: $ \dfrac{1}{y^2 + 17y + 72} = \dfrac{1}{(y + 8)(y + 9)}$ Now we have: $ \dfrac{2}{y + 8}- \dfrac{3}{3(y + 9)}- \dfrac{1}{(y + 8)(y + 9)} $ The least common multiple of the denominators is: $ (y + 8)(y + 9)$ In order to get the first term over $(y + 8)(y + 9)$ , multiply by $\dfrac{3(y + 9)}{3(y + 9)}$ $ \dfrac{2}{y + 8} \times \dfrac{3(y + 9)}{3(y + 9)} = \dfrac{6(y + 9)}{(y + 8)(y + 9)} $ In order to get the second term over $(y + 8)(y + 9)$ , multiply by $\dfrac{y + 8}{y + 8}$ $ \dfrac{3}{3(y + 9)} \times \dfrac{y + 8}{y + 8} = \dfrac{3(y + 8)}{(y + 8)(y + 9)} $ In order to get the third term over $(y + 8)(y + 9)$ , multiply by $\dfrac{3}{3}$ $ \dfrac{1}{(y + 8)(y + 9)} \times \dfrac{3}{3} = \dfrac{3}{(y + 8)(y + 9)} $ Now we have: $ \dfrac{6(y + 9)}{(y + 8)(y + 9)} - \dfrac{3(y + 8)}{(y + 8)(y + 9)} - \dfrac{3}{(y + 8)(y + 9)} $ $ = \dfrac{ 6(y + 9) - 3(y + 8) - 3} {(y + 8)(y + 9)} $ Expand: $ = \dfrac{6y + 54 - 3y - 24 - 3}{3y^2 + 51y + 216} $ $ = \dfrac{3y + 27}{3y^2 + 51y + 216}$ Simplify: $ = \dfrac{y + 9}{y^2 + 17y + 72}$